Page 72 - DMTH403_ABSTRACT_ALGEBRA

Unit 5: Normal Subgroups

Figure 5.1: Geometric Representation of the Generators of D 8 Notes

We can generalise D to the dihedral group
8
D = < { x, y | x = e, y = e, xy = y x} >, for n > 2.
n
2
-1
2n
5.2 Quotient Groups
Here we will use a property of normal subgroups to create a new group. This group is analogous
to the concept of quotient spaces given in the Linear Algebra course.
Let H be a normal subgroup of a group G. Then gH = Hg for every g  G. Consider the collection
of all cosets of H in G. (Note that since H  G, we need not write ‘left coset’ or ‘right coset; simply
‘coset’ is enough.) We denote this set by G/H. Now, for x, y  H, we have
(Hx) (Hy) = H(xH)y, using associativity,

= HHxy, using normality of H,
= Hxy, since HH = H because H is a subgroup.
Now, we define the product of two cosets Hx and Hy and G/H by (Hx)(Hy) = Hxy for all x, y in G.
As this definition seems to depend on the way in which we represent a coset. Let us discuss this
in detail. Suppose C and C are two cosets, say C = Hx and C = Hy. Then C C = Hxy. But C and
1
2
1
1
2
2
1
C can be written in the form Hx and Hy in several ways. So, you may ask : Does C C depend on
2
1
2
the particular way of writing C and C ?
2
1
In other words, if C = Hx = Hx and C = Hy = Hy , then is C C = Hxy or is C C = Hx y ?
2
1 1
2
1
1
1
1
2
1
Actually, we will show you that Hxy = Hx y , that is, the product of cosets is well-defined.
1 1
Since Hx = Hx and Hy = Hy , xx  H, yy  H.
-1
-1
1
1
1
1
-l
-1
-1
-1
 (xy) (x y ) = (xy) (y x ) = x (yy ) x l -1
1
1
1
1
l
= x (yy )x (xx )  H, since xx  H and H  G
-1
-1
-1
1
1
1
i.e:, (xy) (x y )  H.
-1
1 1
 Hxy = Hx y .
1 1
So, we have shown you that multiplication is a well-defined binary operation on G/H.
We will now show that (G/H,.) is a group.
Theorem 5: Let H be a normal subgroup of a group G and G/H denote the set of all cosets of H
in G. Then G/H becomes a group under multiplication defined by Hx . Hy = Hxy, x, y  G. The
coset H = He is the identity of G/H and the inverse of Hx is the coset Hx -1
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